Math 482 (Lecture 6): The simplex method III: lexicographic anticycling The problem Let's assume for now that we can find a bfs to start with (Q3 at the end of Lecture 5). If when we pivot, the b value in pivot row of the first column is 0, the new bfs and only bfs give the same objective value. So, can worry that at the next step, we revisit a previous bfs with the same objective value and the simplex method never terminates because it gets stuck in an infinite loop. This lecture's .pdf note assumes this situation Cycling can actually happen if you made deterministic steps. See Example 2.7 of the textbook. * There are a couple of ways to avoid cycling: lexicography, Bland's rule, randomness. * We discuss lexicography first. See Chapter 14.2 of the textbook. * Randomness as an approach we mostly don't discuss (although it works): just randomly break ties, and with probability 1 you will eventually get out of any cycle. (Still here there's something to think about to ensure you eventually reach a minimum.) Definition:

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